A Mathematical Theory of the Dimensions and Units of Physical Quantities
The aim of this investigation is to disclose the mathematical status of the dimensions and units of physical quantities. Every physicist knows how to perform a dimensional analysis and how to calculate the units of a dimensionally complex magnitude, but no theory proper has hitherto justified these operations. Worse: the two concepts are often mixed up. Even worse: they are frequently “defined” by reference to measurement operations. In this paper an algebra of dimensions and an algebra of units are built with elementary set theoretic and algebraic resources. By supplying unambiguous and exact elucidations of the concepts of dimension and of unit, those theories may contribute to clearing up some of the confusions that becloud the subject.
KeywordsEquivalence Class Abelian Group Equivalence Relation Mathematical Theory Physical System
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